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"Renew Lease" has to do with the way your iPod Touch connects to your grandparent's wifi router, and then the Internet.
Go to Settings, Wi-Fi, and then the Blue > in the circle. This will open and show you the wifi info of your connection to the wifi router.
You will see "DHCP", "BootIP", "Static". The DHCP should be showing you the IP address, Subnet Mask, Router, DNS, Search Domains, Client ID. This info is "leased" to your iPod Touch from your grandparent's wifi router. Clicking the "Renew Lease" will just send a request from your iPod Touch to renew your connection to the wifi router.
Usually, DHCP leases are good for one hour, or one day, or however long the wifi router is configured. Normally, you don't have to renew the lease, but it hurts nothing to touch that button.
Make sure you write down all of the IP address, Subnet Mask, Router, DNS, Search Domains, Client ID, before you do so. I doubt if anything will change.
Do you notice the blue pie slice at the top-left of your screen? It is between the word "iPod" and the displayed time. This shows whether or not you are connected to the internet.
One last thing....
You can reset the wifi connection by resetting the network settings on your iPod Touch by doing this:
Go to Settings, General, Reset, Reset Network Settings. A screen will pop up and "Reset Network Settings" will be shown to you. Touch the Reset Network Settings.
Go back to Settings, Wifi, and let the iPod Touch "find" your grandparent's wifi. Re-connect to your grandparent's wifi and RE-ENTER the password. You should be able to reconnect to the internet.